Bertrand's postulate implies that there are infinitely many
numbers a for which floor(2a),
floor(22a),
floor(222a), ...
are all prime. The smallest of these is
a = 1.25164 75977 90463 01759 44320 53623... and
generates the Bertrand primes: 2, 5, 37,
137438953481, .... The next Bertrand prime has 41,373,247,571
digits. [Caldwell]